Steel and Composite Structures

  • 제33권1호
  • /
  • Pages.133-142
  • /
  • 2019
  • /
  • 1229-9367(pISSN)
  • /
  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Static stability analysis of axially functionally graded tapered micro columns with different boundary conditions

  • Akgoz, Bekir (Department of Civil Engineering, Faculty of Engineering, Akdeniz University)
  • 투고 : 2019.06.22
  • 심사 : 2019.09.05
  • 발행 : 2019.10.10
⟨ 이전 논문 다음 논문 ⟩

초록

In the present study, microstructure-dependent static stability analysis of inhomogeneous tapered micro-columns is performed. It is considered that the micro column is made of functionally graded materials and has a variable cross-section. The material and geometrical properties of micro column vary continuously throughout the axial direction. Euler-Bernoulli beam and modified couple stress theories are used to model the nonhomogeneous micro column with variable cross section. Rayleigh-Ritz solution method is implemented to obtain the critical buckling loads for various parameters. A detailed parametric study is performed to examine the influences of taper ratio, material gradation, length scale parameter, and boundary conditions. The validity of the present results is demonstrated by comparing them with some related results available in the literature. It can be emphasized that the size-dependency on the critical buckling loads is more prominent for bigger length scale parameter-to-thickness ratio and changes in the material gradation and taper ratio affect significantly the values of critical buckling loads.

키워드

과제정보

연구 과제 주관 기관 : Akdeniz University

참고문헌

  1. Aifantis, E.C. (1999), "Gradient deformation models at nano, micro, and macroscales", J. Eng. Mater. Technol., 121, 189-202. https://doi.org/10.1115/1.2812366
  2. Ak, C., Yildiz, A., Akdagli, A. and Bicer, M.B. (2017), "Computing the pull-in voltage of fixed-fixed micro-actuators by artificial neural network", Microsyst. Technol., 23(8), 3537-3546. https://doi.org/10.1007/s00542-016-3128-4
  3. Akgoz, B. and Civalek, O. (2011), "Buckling analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories", J. Comput. Theor. Nanostruct., 8(9), 1821-1827. https://doi.org/10.1166/jctn.2011.1888
  4. Akgoz, B. and Civalek, O. (2013a), "Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity", Struct. Eng. Mech., Int. J., 48(2), 195-205. https://doi.org/10.12989/sem.2013.48.2.195
  5. Akgoz, B. and Civalek, O. (2013b), "Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory", Compos. Struct., 98, 314-322. https://doi.org/10.1016/j.compstruct.2012.11.020
  6. Al-Basyouni, K.S., Tounsi, A. and Mahmoud, S.R. (2015), "Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position", Compos. Struct., 125, 621-630. https://doi.org/10.1016/j.compstruct.2014.12.070
  7. Amar, L.H.H., Kaci, A., Yeghnem, R. and Tounsi, A. (2018), "A new four-unknown refined theory based on modified couple stress theory for size-dependent bending and vibration analysis of functionally graded micro-plate", Steel Compos. Struct., Int. J., 26(1), 89-102. https://doi.org/10.12989/scs.2018.26.1.089
  8. Anandrao, K.S., Gupta, R.K., Ramchandran, P. and Rao, G.V. (2010), "Thermal post-buckling analysis of uniform slender functionally graded material beams", Struct. Eng. Mech., Int. J., 36(5), 545-560. https://doi.org/10.12989/sem.2010.36.5.545
  9. Arefi, M. (2018), "Nonlocal free vibration analysis of a doubly curved piezoelectric nano shell", Steel Compos. Struct., Int. J., 27(4), 479-493. https://doi.org/10.12989/scs.2018.27.4.479
  10. Atmane, H.A., Tounsi, A., Ziane, N. and Mechab, I. (2011), "Mathematical solution for free vibration of sigmoid functionally graded beams with varying cross-section", Steel Compos. Struct., Int. J., 11(6), 489-504. https://doi.org/10.12989/scs.2011.11.6.489
  11. Atmane, H.A., Tounsi, A., Bernard, F. and Mahmoud, S.R. (2015), "A computational shear displacement model for vibrational analysis of functionally graded beams with porosities", Steel Compos. Struct., Int. J., 19(2), 369-384. https://doi.org/10.12989/scs.2015.19.2.369
  12. Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., Int. J., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603
  13. Aydogdu, M. (2008), "Semi-inverse method for vibration and buckling of axially functionally graded beams", J. Reinforced Plast. Comp., 27(7), 683-691. https://doi.org/10.1177/0731684407081369
  14. Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E.A.A. and Mahmoud, S.R. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model", Steel Compos. Struct., Int. J., 18(4), 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063
  15. Bennai, R., Atmane, H.A. and Tounsi, A. (2015), "A new higherorder shear and normal deformation theory for functionally graded sandwich beams", Steel Compos. Struct., Int. J., 19(3), 521-546. https://doi.org/10.12989/scs.2015.19.3.521
  16. Boeing (2019), https://www.boeing.com/commercial/787/bydesign/#/advanced-composite-use
  17. Ebrahimi, F. and Mahmoodi, F. (2019), "A modified couple stress theory for buckling analysis of higher order inhomogeneous microbeams with porosities", Proc. Inst. Mech. Eng. C-J. Mech. Eng. Sci., 233(8), 2855-2866. https://doi.org/10.1177/0954406218791642
  18. Ehyaei, J. and Akbarizadeh, M.R. (2017), "Vibration analysis of micro composite thin beam based on modified couple stress", Struct. Eng. Mech., Int. J., 64(4), 403-411.
  19. Eringen, A.C. (1967), "Theory of micropolarplates", Z. Angew. Math. Phys., 18, 12-30. https://doi.org/10.1007/BF01593891
  20. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10, 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  21. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.332803
  22. Fleck, N.A. and Hutchinson, J.W. (1993), "A phenomenological theory for strain gradient effects in plasticity", J. Mech. Phys. Solids, 41, 1825-1857. https://doi.org/10.1016/0022-5096(93)90072-N
  23. Fleck, N.A. and Hutchinson, J.W. (2001), "A reformulation of strain gradient plasticity", J. Mech. Phys. Solids, 49, 2245-2271. https://doi.org/10.1016/S0022-5096(01)00049-7
  24. Ghayesh, M.H. (2018a), "Mechanics of tapered AFG sheardeformable microbeams", Microsyst. Technol., 24(4), 1743-1754. https://doi.org/10.1007/s00542-018-3764-y
  25. Ghayesh, M.H. (2018b), "Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams", Appl. Math. Model., 59, 583-596. https://doi.org/10.1016/j.apm.2018.02.017
  26. Ghayesh, M.H. (2018c), "Nonlinear vibrations of axially functionally graded Timoshenko tapered beams", J. Comput. Nonlinear Dyn., 13(4), 041002. https://doi.org/10.1115/1.4039191
  27. Ghayesh, M.H. (2018d), "Vibration analysis of shear-deformable AFG imperfect beams", Compos. Struct., 200, 910-920. https://doi.org/10.1016/j.compstruct.2018.03.091
  28. Ghayesh, M.H. (2019), "Resonant dynamics of axially functionally graded imperfect tapered Timoshenko beams", J. Vib. Control, 25(2), 336-350. https://doi.org/10.1177/1077546318777591
  29. Ghayesh, M.H. and Farokhi, H. (2018a), "Bending and vibration analyses of coupled axially functionally graded tapered beams", Nonlinear Dyn., 91(1), 17-28. https://doi.org/10.1007/s11071-017-3783-8
  30. Ghayesh, M.H. and Farokhi, H. (2018b), "Mechanics of tapered axially functionally graded shallow arches", Compos. Struct., 188, 233-241. https://doi.org/10.1016/j.compstruct.2017.11.017
  31. Ghayesh, M.H., Farokhi, H., Gholipour, A. and Tavallaeinejad, M. (2017), "Nonlinear bending and forced vibrations of axially functionally graded tapered microbeams", Int. J. Eng. Sci., 120, 51-62. https://doi.org/10.1016/j.ijengsci.2017.03.010
  32. Gusso, A., Viana, R.L., Mathias, A.C. and Caldas, I.L. (2019), "Nonlinear dynamics and chaos in micro/ nanoelectromechanical beam resonators actuated by two-sided electrodes", Chaos Soliton Fract., 122, 6-16. https://doi.org/10.1016/j.chaos.2019.03.004
  33. Hadi, A., Nejad, M.Z., Rastgoo, A. and Hosseini, M. (2018), "Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory", Steel Compos. Struct., Int. J., 26(6), 663-672. https://doi.org/10.12989/scs.2018.26.6.663
  34. Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., Int. J., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  35. Hamzehkolaei, N.S., Malekzadeh, P. and Vaseghi, J. (2011), "Thermal effect on axisymmetric bending of functionally graded circular and annular plates using DQM", Steel Compos. Struct., Int. J., 11(4), 341-358. https://doi.org/10.12989/scs.2011.11.4.341
  36. He, X.J., Wu, Q., Wang, Y., Song, M.X. and Yin, J.H. (2009), "Numerical simulation and analysis of electrically actuated microbeam-based MEMS capacitive switch", Microsyst. Technol., 15(2), 301-307. https://doi.org/10.1007/s00542-008-0702-4
  37. Houari, M.S.A., Bessaim, A., Bernard, F., Tounsi, A. and Mahmoud, S.R. (2018), "Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter", Steel Compos. Struct., Int. J., 28(1), 13-24. https://doi.org/10.12989/scs.2018.28.1.013
  38. Jahangiri, R., Jahangiri, H. and Khezerloo, H. (2015), "FGM micro-gripper under electrostatic and intermolecular Van-der Waals forces using modified couple stress theory", Steel Compos. Struct., Int. J., 18(6), 1541-1555. https://doi.org/10.12989/scs.2015.18.6.1541
  39. Jia, X.L., Ke, L.L., Zhong, X.L., Sun, Y., Yang, J. and Kitipornchai, S. (2018), "Thermal-mechanical-electrical buckling behavior of functionally graded micro-beams based on modified couple stress theory", Compos. Struct., 202, 625-634. https://doi.org/10.1016/j.compstruct.2018.03.025
  40. Karami, B., Janghorban, M. and Tounsi, A. (2018), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., Int. J., 27(2), 201-216. https://doi.org/10.12989/scs.2018.27.2.201
  41. Khaniki, H.B. and Rajasekaran, S. (2018), "Mechanical analysis of non-uniform bi-directional functionally graded intelligent microbeams using modified couple stress theory", Mater. Res. Express., 5(5), 055703. https://doi.org/10.1088/2053-1591/aabe62
  42. Khorshidi, M.A., Shariati, M. and Emam, S.A. (2016), "Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory", Int. J. Mech. Sci., 110, 160-169. https://doi.org/10.1016/j.ijmecsci.2016.03.006
  43. Kiani, Y. and Eslami, M.R. (2010), "Thermal buckling analysis of functionally graded material beams", Int. J. Mech. Mater. Des., 6(3), 229-238. https://doi.org/10.1007/s10999-010-9132-4
  44. Kocaturk, T. and Akbas, S.D. (2013), "Thermal post-buckling analysis of functionally graded beams with temperaturedependent physical properties", Steel Compos. Struct., Int. J., 15(5), 481-505. https://doi.org/10.12989/scs.2013.15.5.481
  45. Koiter, W.T. (1964), "Couple stresses in the theory of elasticity", I and II. Proc. K. Ned. Akad. Wet. B, 67, 17-44.
  46. Kong, S., Zhou, S., Nie, Z. and Wang, K. (2009), "Static and dynamic analysis of micro beams based on strain gradient elasticity theory", Int. J. Eng. Sci., 47, 487-498. https://doi.org/10.1016/j.ijengsci.2008.08.008
  47. Lam, D.C.C. and Chong, A.C.M. (2001), "Model and experiments on strain gradient hardening in metallic glass", Mat. Sci. Eng. AStruct., 318(1-2), 313-319. https://doi.org/10.1016/S0921-5093(01)01329-6
  48. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids., 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  49. Lee, H.L. and Chang, W.J. (2009), "Effects of damping on the vibration frequency of atomic force microscope cantilevers using the Timoshenko beam model", Jpn. J. Appl. Phys., 48(6), 065005. https://doi.org/10.1143/JJAP.48.065005
  50. Lei, J., He, Y.M., Guo, S., Li, Z.K. and Liu, D.B. (2016), "Sizedependent vibration of nickel cantilever microbeams: Experiment and gradient elasticity", Aip. Adv., 6(10), 105202. https://doi.org/10.1063/1.4964660
  51. Li, B., Tang, X.S., Xie, H.M. and Xin, Z. (2004), "Strain analysis in MEMS/NEMS structures and devices by using focused ion beam system", Sensor Actuator A-Phys., 111(1), 57-62. https://doi.org/10.1016/j.sna.2003.07.014
  52. Li, S.R., Zhang, J.H. and Zhao, Y.G. (2006), "Thermal postbuckling of functionally graded material Timoshenko beams", Appl. Math. Mech., 27(6), 803-810. https://doi.org/10.1007/s10483-006-0611-y
  53. Li, Z.K., He, Y.M., Lei, J., Guo, S., Liu, D.B. and Wang, L. (2018), "A standard experimental method for determining the material length scale based on modified couple stress theory", Int. J. Mech. Sci., 141, 198-205. https://doi.org/10.1016/j.ijmecsci.2018.03.035
  54. Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids., 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001
  55. Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructuredependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solids., 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007
  56. Ma, H.M., Gao, X.L. and Reddy, J.N. (2010), "A nonclassical Reddy-Levinson beam model based on a modified couple stress theory", Int. J. Multiscale Comput. Eng., 8(2), 167-180. https://doi.org/10.1615/IntJMultCompEng.v8.i2.30
  57. Mahmoud, S.R. and Tounsi, A. (2017), "A new shear deformation plate theory with stretching effect for buckling analysis of functionally graded sandwich plates", Steel Compos. Struct., Int. J., 24(5), 569-578. https://doi.org/10.12989/scs.2017.24.5.569
  58. Marques, L., da Silva, L.S. and Rebelo, C. (2014), "Rayleigh-Ritz procedure for determination of the critical load of tapered columns", Steel Compos. Struct., Int. J., 16(1), 47-60. https://doi.org/10.12989/scs.2014.16.1.047
  59. Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couplestresses in linear elasticity", Arch. Ration. Mech. Anal., 11, 415-448. https://doi.org/10.1007/BF00253946
  60. Nateghi, A., Salamat-talab, M., Rezapour, J. and Daneshian, B. (2012), "Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory", Appl. Math. Model., 36(10), 4971-4987. https://doi.org/10.1016/j.apm.2011.12.035
  61. Nazemnezhad, R. and Kamali, K. (2018), "Free axial vibration analysis of axially functionally graded thick nanorods using nonlocal Bishop's theory", Steel Compos. Struct., Int. J., 28(6), 749-758. https://doi.org/10.12989/scs.2018.28.6.749
  62. Nguyen, D.K. and Tran, T.T. (2018), "Free vibration of tapered BFGM beams using an efficient shear deformable finite element model", Steel Compos. Struct., Int. J., 29(3), 363-377. https://doi.org/10.12989/scs.2018.29.3.363
  63. Nguyen, N.T., Kim, N.I. and Lee, J. (2014), "Analytical solutions for bending of transversely or axially FG nonlocal beams", Steel Compos. Struct., Int. J., 17(5), 639-663. https://doi.org/10.12989/scs.2014.17.5.641
  64. Nguyen, K.T., Thai, T.H. and Vo, T.P. (2015), "A refined higherorder shear deformation theory for bending, vibration and buckling analysis of functionally graded sandwich plates", Steel Compos. Struct., Int. J., 18(1), 91-120. https://doi.org/10.12989/scs.2015.18.1.091
  65. Park, S.K. and Gao, X.L. (2006), "Bernoulli-Euler beam model based on a modified couple stress theory", J. Micromech. Microeng., 16(11), 2355-2359. https://doi.org/10.1088/0960-1317/16/11/015
  66. Park, W.T., Han, S.C., Jung, W.Y. and Lee, W.H. (2016), "Dynamic instability analysis for S-FGM plates embedded in Pasternak elastic medium using the modified couple stress theory", Steel Compos. Struct., Int. J., 22(6), 1239-1259. https://doi.org/10.12989/scs.2016.22.6.1239
  67. Payam, A.F. and Fathipour, M. (2009), "Modeling and dynamic analysis of atomic force microscope based on Euler-Bernoulli beam theory", Dig. J. Nanomater. Bios., 4(3), 565-578.
  68. Pradhan, K.K. and Chakraverty, S. (2013), "Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method", Compos. Part B-Eng., 51, 175-184. https://doi.org/10.1016/j.compositesb.2013.02.027
  69. Rahmani, O., Deyhim, S. and Hosseini, S.A.H. (2018a), "Size dependent bending analysis of micro/nano sandwich structures based on a nonlocal high order theory", Steel Compos. Struct., Int. J., 27(3), 371-388. https://doi.org/10.12989/scs.2018.27.3.371
  70. Rahmani, O., Hosseini, S.A.H., Ghoytasi, I. and Golmohammadi, H. (2018b), "Free vibration of deep curved FG nano-beam based on modified couple stress theory", Steel Compos. Struct., Int. J., 26(5), 607-620. https://doi.org/10.12989/scs.2018.26.5.607
  71. Rezaiee-Pajand, M., Masoodi, A.R. and Alepaighambar, A. (2018), "Lateral-torsional buckling of functionally graded tapered Ibeams considering lateral bracing", Steel Compos. Struct., Int. J., 28(4), 403-414. https://doi.org/10.12989/scs.2018.28.4.403
  72. Sahraee, S. and Saidi, A.R. (2009), "Free vibration and buckling analysis of functionally graded deep beam-columns on twoparameter elastic foundations using the differential quadrature method", Proc. Inst. Mech. Eng. C-J. Mech. Eng. Sci., 223(6), 1273-1284. https://doi.org/10.1243/09544062JMES1349
  73. Saidi, H., Houari, M.S.A., Tounsi, A. and Bedia, E.A. (2013), "Thermo-mechanical bending response with stretching effect of functionally graded sandwich plates using a novel shear deformation theory", Steel Compos. Struct., Int. J., 15(2), 221-245. https://doi.org/10.12989/scs.2013.15.2.221
  74. Salamat-Talab, M., Nateghi, A. and Torabi, J. (2012), "Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory", Int. J. Mech. Sci., 57(1), 63-73. https://doi.org/10.1016/j.ijmecsci.2012.02.004
  75. Shafiei, N. and Kazemi, M. (2017), "Buckling analysis on the bidimensional functionally graded porous tapered nano-/microscale beams", Aerosp. Sci. Technol., 66, 1-11. https://doi.org/10.1016/j.ast.2017.02.019
  76. Shafiei, N. and She, G.L. (2018), "On vibration of functionally graded nano-tubes in the thermal environment", Int. J. Eng. Sci., 133, 84-98. https://doi.org/10.1016/j.ijengsci.2018.08.004
  77. Shafiei, N., Kazemi, M. and Ghadiri, M. (2016a), "Comparison of modeling of the rotating tapered axially functionally graded Timoshenko and Euler-Bernoulli microbeams", Physica E, 83, 74-87. https://doi.org/10.1016/j.physe.2016.04.011
  78. Shafiei, N., Kazemi, M. and Ghadiri, M. (2016b), "Nonlinear vibration of axially functionally graded tapered microbeams", Int. J. Eng. Sci., 102, 12-26. https://doi.org/10.1016/j.ijengsci.2016.02.007
  79. Shafiei, N., Mousavi, A. and Ghadiri, M. (2016c), "On sizedependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams", Int. J. Eng. Sci., 106, 42-56. https://doi.org/10.1016/j.ijengsci.2016.05.007
  80. Shafiei, N., Kazemi, M. and Fatahi, L. (2017), "Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method", Mech. Adv. Mater. Struct., 24(3), 240-252. https://doi.org/10.1080/15376494.2015.1128025
  81. Shafiei, N., Ghadiri, M. and Mahinzare, M. (2019), "Flapwise bending vibration analysis of rotary tapered functionally graded nanobeam in thermal environment", Mech. Adv. Mater. Struct., 26(2), 139-155. https://doi.org/10.1080/15376494.2017.1365982
  82. She, G.L., Yuan, F.G. and Ren, Y.R. (2017), "Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory", Appl. Math. Model., 47, 340-357. https://doi.org/10.1016/j.apm.2017.03.014
  83. She, G.L., Yan, K.M., Zhang, Y.L., Liu, H.B. and Ren, Y.R. (2018), "Wave propagation of functionally graded porous nanobeams based on non-local strain gradient theory", Eur. Phys. J. Plus, 133, 368. https://doi.org/10.1140/epjp/i2018-12196-5
  84. Simsek, M. (2011), "Forced vibration of an embedded singlewalled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory", Steel Compos. Struct., Int. J., 11(1), 59-76. https://doi.org/10.12989/scs.2011.11.1.059
  85. Simsek, M. (2012), "Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods", Comp. Mater. Sci., 61, 257-265. https://doi.org/10.1016/j.commatsci.2012.04.001
  86. Simsek, M. and Reddy, J.N. (2013), "A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory", Compos. Struct., 101, 47-58. https://doi.org/10.1016/j.compstruct.2013.01.017
  87. Tagrara, S.H., Benachour, A., Bouiadjra, M.B. and Tounsi, A. (2015), "On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams", Steel Compos. Struct., Int. J., 19(5), 1259-1277. https://doi.org/10.12989/scs.2015.19.5.1259
  88. Thanh, C.L., Tran, L.V., Vu-Huu, T. and Abdel-Wahab, M. (2019), "The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis", Comput. Method. Appl. Mech., 350, 337-361. https://doi.org/10.1016/j.cma.2019.02.028
  89. Toupin, R.A. (1964), "Theories of elasticity with couple-stresses", Arch. Ration. Mech. Anal., 17, 85-112. https://doi.org/10.1007/BF00253050
  90. Vardoulakis, I. and Sulem, J. (1995), Bifurcation Analysis in Geomechanics, CRC Press.
  91. Wang, C.M., Wang, C.Y. and Reddy, J.N. (2005), Exact Solutions for Buckling of Structural Members, CRC Press.
  92. Wattanasakulpong, N., Prusty, B.G. and Kelly, D.W. (2011), "Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams", Int. J. Mech. Sci., 53(9), 734-743. https://doi.org/10.1016/j.ijmecsci.2011.06.005
  93. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39, 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  94. Younis, M.I., Abdel-Rahman, E.M. and Nayfeh, A. (2003), "A reduced-order model for electrically actuated microbeam-based MEMS", J. Microelectromech. Sci, 12(5), 672-680. https://doi.org/10.1109/JMEMS.2003.818069